MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid - - PowerPoint PPT Presentation

MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Muhammad Mohid October 30th, 2020 Warm-up question: True or false? Suppose A and B are two sets, f : A → B and g : B → A functions. If, g(f (a)) = a for every a ∈ A, then f is invertible and f −1 = g.

What can we actually conclude?

What can we actually conclude?

1 f is injective;

What can we actually conclude?

1 f is injective; 2 f is surjective;

What can we actually conclude?

1 f is injective; 2 f is surjective; 3 g is injective;

What can we actually conclude?

1 f is injective; 2 f is surjective; 3 g is injective; 4 g is surjective.

What is wrong with this proof?

Theorem

Suppose that I ⊆ R is an interval and f : I → I is an invertible and differentiable function, with f −1 = f and f ′(x) = 0 for every x ∈ I. Then, for every x ∈ I, we have f ′(x) = 1 or f ′(x) = −1.

What is wrong with this proof?

Theorem

Suppose that I ⊆ R is an interval and f : I → I is an invertible and differentiable function, with f −1 = f and f ′(x) = 0 for every x ∈ I. Then, for every x ∈ I, we have f ′(x) = 1 or f ′(x) = −1. Examples: I = R and f (x) = x, or f (x) = −x.

Proof.

Since f ′(x) = 0 for every x ∈ I, I can use the formula f ′ =

• f −1′ = 1

f ′ . Now fix x ∈ I: using again that f ′(x) = 0, I can multiply both sides of the equation by f ′(x) and find that

f ′(x) 2 = 1, so f ′(x) = ±1.

Worm up

A worm is crawling accross the table. The path of the worm looks something like this:

True or False?

The position of the worm is a function.

Worm function

A worm is crawling accross the table. For any time t, let f (t) be the position of the worm. This defines a function f .

1 What is the domain of f ? 2 What is the codomain of f ? 3 What is the range of f ?

Before next class...

• Watch videos 4.5, 4.7, 4.8 and 4.9.
• Download the next class’s slides (no need to look at them!)